Since the set of users served by each slice is uncertain in long time-scales, we need to take an expectation with respect to \(\mathcal {K}_s\)  from (REF )–(REF ) and (REF ).
The argument of revenue functions \(\phi _{k,S_n}(\cdot )\)  and \(\phi _{k,L_n}(\cdot )\)  in (REF ) and (REF ), which is \(\min ( r_k,d_k)\) , is a concave function of \(r_k\) . Since \(\phi _{k,S_n}(\cdot )\)  and \(\phi _{k,L_n}(\cdot )\)  are both non-decreasing concave functions, based on the rule for function compositions in [1]}, revenue functions are also concave. After taking expectations, (REF ) and (REF ) remain concave functions.
